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Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations
Blow-up set for a superlinear heat equation and pointedness of the initial data
1. | Division of Mathematical Science, Department of Systems Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikaneyama-cho, Toyonaka 560-8531, Japan |
References:
[1] |
X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
doi: 10.1016/0022-0396(89)90081-8. |
[2] |
T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential, J. Differential Equations, 244 (2008), 766-802.
doi: 10.1016/j.jde.2007.11.004. |
[3] |
C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, J. Math. Anal. Appl., 335 (2007), 418-427.
doi: 10.1016/j.jmaa.2007.01.079. |
[4] |
A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion, SIAM J. Math. Anal., 18 (1987), 711-721.
doi: 10.1137/0518054. |
[5] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
doi: 10.1512/iumj.1985.34.34025. |
[6] |
Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786. |
[7] |
Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations, 249 (2010), 1056-1077.
doi: 10.1016/j.jde.2010.03.028. |
[8] |
Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$, J. Differential Equations, 250 (2011), 2508-2543.
doi: 10.1016/j.jde.2010.12.008. |
[9] |
Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II, J. Differential Equations, 252 (2012), 1835-1861.
doi: 10.1016/j.jde.2011.08.040. |
[10] |
Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiana Univ. Math. J., 61 (2012), 627-663.
doi: 10.1512/iumj.2012.61.4596. |
[11] |
Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation, Ann. Inst. H. Poincaré Anal., 31 (2014), 231-247.
doi: 10.1016/j.anihpc.2013.03.001. |
[12] |
Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.
doi: 10.1002/cpa.3160420607. |
[13] |
K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion, Adv. Differential Equations, 7 (2002), 1003-1024. |
[14] |
K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion, Math. Ann., 327 (2003), 487-511.
doi: 10.1007/s00208-003-0463-4. |
[15] |
K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128.
doi: 10.1016/j.jde.2004.10.021. |
[16] |
N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation, Indiana Univ. Math. J., 50 (2001), 591-610.
doi: 10.1512/iumj.2001.50.1905. |
[17] |
N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion, J. Math. Anal. Appl., 261 (2001), 350-368.
doi: 10.1006/jmaa.2001.7530. |
[18] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.
doi: 10.1007/978-3-7643-8442-5. |
[19] |
S. Sato, Life span of solutions with large initial data for a superlinear heat equation, J. Math. Anal. Appl. 343 (2008), 1061-1074.
doi: 10.1016/j.jmaa.2008.02.018. |
[20] |
J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596.
doi: 10.1080/03605309208820896. |
[21] |
J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J., 42 (1993), 445-476.
doi: 10.1512/iumj.1993.42.42021. |
[22] |
F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224.
doi: 10.1016/0022-0396(84)90081-0. |
[23] |
H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. Math. Soc. Japan, 56 (2004), 993-1005.
doi: 10.2969/jmsj/1190905445. |
[24] |
H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation, J. Math. Soc. Japan, 56 (2004), 1007-1017.
doi: 10.2969/jmsj/1190905446. |
[25] |
H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505-542.
doi: 10.1016/S0294-1449(01)00088-9. |
[26] |
H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549.
doi: 10.1007/s002200100589. |
[27] |
H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations, Mathematics mathematics education (Bethlehem, 2000), 337-347, World Sci. Publ., River Edge, NJ, 2002. |
[28] |
H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J., 133 (2006), 499-525.
doi: 10.1215/S0012-7094-06-13333-1. |
show all references
References:
[1] |
X. Y. Chen and H. Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations, 78 (1989), 160-190.
doi: 10.1016/0022-0396(89)90081-8. |
[2] |
T. Cheng and G. F. Zheng, Some blow-up problems for a semilinear parabolic equation with a potential, J. Differential Equations, 244 (2008), 766-802.
doi: 10.1016/j.jde.2007.11.004. |
[3] |
C. Cortazar, M. Elgueta and J. D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, J. Math. Anal. Appl., 335 (2007), 418-427.
doi: 10.1016/j.jmaa.2007.01.079. |
[4] |
A. Friedman and A. A. Lacey, The blow-up time for solutions of nonlinear heat equations with small diffusion, SIAM J. Math. Anal., 18 (1987), 711-721.
doi: 10.1137/0518054. |
[5] |
A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J., 34 (1985), 425-447.
doi: 10.1512/iumj.1985.34.34025. |
[6] |
Y. Fujishima, Location of the blow-up set for a superlinear heat equation with small diffusion, Differential Integral Equations, 25 (2012), 759-786. |
[7] |
Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation with small diffusion, J. Differential Equations, 249 (2010), 1056-1077.
doi: 10.1016/j.jde.2010.03.028. |
[8] |
Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$, J. Differential Equations, 250 (2011), 2508-2543.
doi: 10.1016/j.jde.2010.12.008. |
[9] |
Y. Fujishima and K. Ishige, Blow-up for a semilinear parabolic equation with large diffusion on $R^N$. II, J. Differential Equations, 252 (2012), 1835-1861.
doi: 10.1016/j.jde.2011.08.040. |
[10] |
Y. Fujishima and K. Ishige, Blow-up set for a semilinear heat equation and pointedness of the initial data, Indiana Univ. Math. J., 61 (2012), 627-663.
doi: 10.1512/iumj.2012.61.4596. |
[11] |
Y. Fujishima and K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation, Ann. Inst. H. Poincaré Anal., 31 (2014), 231-247.
doi: 10.1016/j.anihpc.2013.03.001. |
[12] |
Y. Giga and R. V. Kohn, Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., 42 (1989), 845-884.
doi: 10.1002/cpa.3160420607. |
[13] |
K. Ishige, Blow-up time and blow-up set of the solutions for semilinear heat equations with large diffusion, Adv. Differential Equations, 7 (2002), 1003-1024. |
[14] |
K. Ishige and N. Mizoguchi, Location of blow-up set for a semilinear parabolic equation with large diffusion, Math. Ann., 327 (2003), 487-511.
doi: 10.1007/s00208-003-0463-4. |
[15] |
K. Ishige and H. Yagisita, Blow-up problems for a semilinear heat equation with large diffusion, J. Differential Equations, 212 (2005), 114-128.
doi: 10.1016/j.jde.2004.10.021. |
[16] |
N. Mizoguchi and E. Yanagida, Life span of solutions with large initial data in a semilinear parabolic equation, Indiana Univ. Math. J., 50 (2001), 591-610.
doi: 10.1512/iumj.2001.50.1905. |
[17] |
N. Mizoguchi and E. Yanagida, Life span of solutions for a semilinear parabolic problem with small diffusion, J. Math. Anal. Appl., 261 (2001), 350-368.
doi: 10.1006/jmaa.2001.7530. |
[18] |
P. Quittner and P. Souplet, Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.
doi: 10.1007/978-3-7643-8442-5. |
[19] |
S. Sato, Life span of solutions with large initial data for a superlinear heat equation, J. Math. Anal. Appl. 343 (2008), 1061-1074.
doi: 10.1016/j.jmaa.2008.02.018. |
[20] |
J. J. L. Velázquez, Higher-dimensional blow-up for semilinear parabolic equations, Comm. Partial Differential Equations, 17 (1992), 1567-1596.
doi: 10.1080/03605309208820896. |
[21] |
J. J. L. Velázquez, Estimates on the $(n-1)$-dimensional Hausdorff measure of the blow-up set for a semilinear heat equation, Indiana Univ. Math. J., 42 (1993), 445-476.
doi: 10.1512/iumj.1993.42.42021. |
[22] |
F. B. Weissler, Single point blow-up for a semilinear initial value problem, J. Differential Equations 55 (1984), 204-224.
doi: 10.1016/0022-0396(84)90081-0. |
[23] |
H. Yagisita, Blow-up profile of a solution for a nonlinear heat equation with small diffusion, J. Math. Soc. Japan, 56 (2004), 993-1005.
doi: 10.2969/jmsj/1190905445. |
[24] |
H. Yagisita, Variable instability of a constant blow-up solution in a nonlinear heat equation, J. Math. Soc. Japan, 56 (2004), 1007-1017.
doi: 10.2969/jmsj/1190905446. |
[25] |
H. Zaag, On the regularity of the blow-up set for semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 19 (2002), 505-542.
doi: 10.1016/S0294-1449(01)00088-9. |
[26] |
H. Zaag, One-dimensional behavior of singular $N$-dimensional solutions of semilinear heat equations, Comm. Math. Phys., 225 (2002), 523-549.
doi: 10.1007/s002200100589. |
[27] |
H. Zaag, Regularity of the blow-up set and singular behavior for semilinear heat equations, Mathematics mathematics education (Bethlehem, 2000), 337-347, World Sci. Publ., River Edge, NJ, 2002. |
[28] |
H. Zaag, Determination of the curvature of the blow-up set and refined singular behavior for a semilinear heat equation, Duke Math. J., 133 (2006), 499-525.
doi: 10.1215/S0012-7094-06-13333-1. |
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